Double heavy tri-hadron bound state via delocalized  $\pi$ bond

Abstract：

The number of exotic candidates that are beyond the conventional quark model has increased dramatically over recent decades. Some of these can be viewed as analogues of the deuteron. Similarly, the existence of the triton indicates that bound states formed by three hadrons could also exist.  To illustrate this possibility, we study the $DD^*K$ and $BB^*\bar{K}$ systems using the Born-Oppenheimer approximation. To leading order, only one-pion exchange potentials are considered. This means that the three constituents share one virtual pion. This is similar to the role of the delocalized {\it $\pi$ bond} for the formation of benzene in chemistry.  After solving the Schr\"odinger equation, we find two three-body $DD^*K$ and $BB^*\bar{K}$ bound states with masses $4317.92_{-4.32}^{+3.66}~\mathrm{MeV}$ and $11013.65_{-8.84}^{+8.49}~\mathrm{MeV}$, respectively. The masses of their $D\bar{D}^*K$ and $B\bar{B}^*\bar{K}$ analogues are $4317.92_{-6.55}^{+6.13}~\mathrm{MeV}$ and $11013.65_{-9.02}^{+8.68}~\mathrm{MeV}$, respectively. From the experimental side, the $D\bar{D}^*K$ bound state could be found by analyzing the current world data on the $B\to J/\psi\pi\pi K$ process, by focusing on the $J/\psi \pi K$ channel.